we get = (cos A) + (sin A) = cos A + sin A = ( ) This is true for all A such that ° A °. So, this is a trigonometric identity. Let us now divide ( ) by AB . We get or, = + tan A = sec A ( ) Is this equation true for A = °?
Yes, it is. What about A = °? Well, tan A and sec A are not defined for A = °. So, ( ) is true for all A such that ° A °.
Let us see what we get on dividing ( ) by BC . We get Fig. . = cot A + = cosec A ( ) Note that cosec A and cot A are not defined for A = °.
Therefore ( ) is true for all A such that ° < A °. Using these identities, we can express each trigonometric ratio in terms of other trigonometric ratios, i.e., if any one of the ratios is known, we can also determine the values of other trigonometric ratios. Let us see how we can do this using these identities. Suppose we know that tan A = Then, cot A = .
Since, sec A = + tan A = , sec A = , and cos A = Again, sin A = . Therefore, cosec A = . Example : Express the ratios cos A, tan A and sec A in terms of sin A. Solution : Since cos A + sin A = , therefore, cos A = – sin A, i.e., cos A = This gives cos A = Hence, tan A = sin A cos A = and sec A = cos A – sin A Example : Prove that sec A ( – sin A)(sec A + tan A) = .